4 1. Various Ways of Representing Surfaces and Examples
Figure 1.3. A torus and a handle.
Another familiar example of a surface is a torus—just as the
sphere is the surface of an idealised ball, the torus is the surface of
an idealised doughnut (or perhaps a bagel, depending on what sort
of diet one is on). Like our first three examples, it is a surface of
revolution, and may be obtained by rotating a circle around a line
which lies in the plane of the circle, but does not intersect it. We will
derive a nice equation (1.5) for the torus in the next lecture.
We can obtain new surfaces with qualitatively distinctive shapes
by the procedure called “attaching a handle”. A handle can be
thought of as a torus with a hole (or if you like, an inner tube with
a small patch cut out), as shown in Figure 1.3—this is attached to
a hole cut in a given surface. Applying this procedure to a sphere
produces a surface in the general shape of a torus. If we continue to
attach more handles, we obtain something reminiscent of a pretzel
with an increasing number of holes or, alternatively, a chain of tori
linked to each other—Figure 1.4 shows a sphere with two handles.
Like all the surfaces we have dealt with so far, these surfaces can
also be represented by equations with a certain amount of effort (see
b. Equations vs. other methods. We have obtained several dif-
ferent surfaces as the set of points whose coordinates (x, y, z) satisfy
one equation or another. It is natural to ask what sort of equations
will always yield nice, recognisable surfaces. Will any old equation
do? Or must we impose some restrictions? And conversely, can we
represent every surface by an equation?